Mills College Tuning Digest # 1577

Topic No. 7

Date: Mon, 9 Nov 1998 21:03:19 -0800 (PST)
From: "M. Schulter"
To: Tuning Digest
Subject: Re: Vicentino 24 tuning for two keyboards
Message-ID:

Nicola Vicentino (1511-1576) is a figure of great interest to xenharmonicists, and in this post I'd like to focus especially on one approach to putting his theory of the archicembalo and archiorgano into practice on 20th-century microtunable synthesizers.

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1. Practice and theory
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Like the sophisticated Gothic theorists of the era 1200-1435 or so who adapted Pythagorean traditions to complex techniques of polyphony, Vicentino sought to adapt the traditions of Ptolemy and Aristoxenos to the requirements of 16th-century polyphony. Also, like some of his Gothic counterparts (e.g. Prosdocimus, 1413), he found that 12 notes per octave were not enough.

Recognizing a keyboard temperament of 1/4-comma meantone with its pure major thirds, or some approximation, as established "modern" practice, Vicentino in fact defines the prevalent contemporary style as "mixed and tempered music." Building on the usual 12-note meantone scale, and seeking to implement the Greek diatonic, chromatic, and enharmonic genera on an archicembalo or "superharpsichord" as we might now say, he proposes an instrument with 36 or 38 notes per octave.

For this instrument he proposes two alternate tunings, one of them providing a complete system of all three genera which seems essentially the same as 31-tone equal temperament (31-tet), plus either five or seven extra notes supplying pure fifths for some or all of the basic diatonic notes.

The second tuning ideally uses 38 notes per octave: 19 notes to implement the diatonic and chromatic genera, and another 19 to provide pure fifths for these notes. Thus either system, while premised on 1/4-comma meantone or 31-tet with its somewhat narrow or "blunted" fifths, can provide just fifths (and minor thirds) for at least some commonly used sonorities which Zarlino (1558) describes as the harmonia perfetta of third and fifth above the bass, and Lippius (1612) calls the trias harmonica or "triad."

Bill Alves has done a fine article on Vicentino's modified just intonation system for extended keyboards.[1] Additionally, a complete English edition of his famous treatise of 1555 by Maria Rika Maniates and Claude Palisca has now been published.[2] These sources may lend some perspective to what follows.

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2. Vicentino 24: a subset
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At the end of the 20th century, one approach to a partial implementation of Vicentino's system is to use two standard 12-note keyboards, yielding a 24-note subset of his archicembalo tunings.

As it happens, his two alternative tunings for 36 notes seem to share 24 notes in common: the 19 diatonic and chromatic notes of 1/4-comma meantone from Gb to B#; and five additional notes providing pure rather than usual meantone fifths above G, A, C, D, and E. These five notes are finals of the five natural modes most commonly used in Renaissance music (D Dorian, E Phrygian, G Mixolydian, A Aeolian, and C Ionian). Thus the availability of the just fifths D, E, G, A, and B makes possible a pure harmonia perfetta on the finals of these modes, or as Lippius might say, a pure triad.

With Vicentino's instrument, the first 19 notes are arranged into three "ranks." The first two ranks are equivalent to the white and black keys of a usual 12-note meantone keyboard, while the third rank supplies seven additional accidentals:

Rank 3: b#  db   d#     e#  gb    ab   a#     b#   
Rank 2:    c#      eb      f#    g#     bb
Rank 1:  c     d     e   f     g     a     b   c

The additional five notes providing pure fifths to c, d, e, g, and a might be notated g5, a5, b5, d5, and e5. These notes apparently make up the "sixth rank" of Vicentino's first tuning[3] (where the fourth and fifth ranks provide the notes of the enharmonic genus, comprising with the first three ranks a 31-note set essentially equivalent to 31-tet).

Thus our "Vicentino 24" subset looks like this:

Rank 6:  c5    d5    e5        g5    a5    b5  c5  
Rank 3: b#  db   d#     e#  gb    ab   a#     b#   
Rank 2:    c#      eb      f#    g#     bb
Rank 1:  c     d     e   f     g     a     b   c

The tones of Rank 6 (e.g. c5) are higher than their equivalents in Rank 1 (e.g. c) by about 5.38 cents, the amount by which the fifth is tempered in 1/4-comma meantone to obtain pure major thirds. This difference is one of the senses in which Vicentino uses the term "comma," a term with other interpretations to make life a bit more interesting (and complicated) for modern readers and analysts.

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3. Implementation on two keyboards
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The "Vicentino 24" tuning which follows might be implemented either on an acoustical instrument with two manuals (e.g. a harpsichord), or on a synthesizer system of some kind supporting independent tunings for two keyboards. Each keyboard might be supported by its own synthesizer, or a multitimbral synthesizer such as the Yamaha TX-802 might support both keyboards through its "part-tuning" feature.

While 31-tet is the ideal tuning for Vicentino's complete scheme, since it permits a closed system with the equal fivefold division of the whole-tone he describes, 1/4-comma meantone should give essentially the same results with our Vicentino 24 subset.

Generating the first 19 notes of this subset, corresponding to Vicentino's first three ranks, can be accomplished on synthesizers such as the TX-802 by selecting two transpositions of 1/4-comma meantone (Yamaha's "MeanTone" preset). The first keyboard, in a standard Eb-G# tuning with a "Wolf" fourth or fifth between these notes, represents the 12 notes of his first two ranks, the seven diatonic notes plus the five usual accidentals (c#, eb, f#, g#, bb).

The second keyboard, in a G-B# tuning, adds the seven accidentals of the third rank -- five of these accidentals mapped to the black keys (db, d#, gb, ab, a#), plus e# and b# on the usual f and c keys.

On the TX-802, for example, the first Eb-G# keyboard would be selected on the front panel as "MeanTone" with Yamaha's "key of C" specified; the second G-B# keyboard is Yamaha's "key of E." A more general hint for the TX-802's preset Pythagorean and meantone tunings: take the Yamaha "key" and add an ascending minor third to find the lower note of the Wolf fourth. Thus "key of F" means a Wolf at Ab-C#, and "key of G" a Wolf at Bb-D#, etc.

While simply selecting these two preset 1/4-comma meantone temperaments supplies 19 notes of our Vicentino 24 subset, we need to custom-tune the remaining five notes of the second keyboard to get our desired pure fifths. With synthesizer keyboards, the best approach might be to tune these notes (c5', e5', g5', a5', b5') to the nearest approximation of a perfect fifth above f, a, c', d', and e' on the lower keyboard. This would call for intervals of 599 tuning steps on a 1024-tet synthesizer, 449 steps in 768-tet, etc.[4]

Here is a diagram showing the notes on the two keyboards in cents, with c' as the point of reference:

      117.1   269.21             620.53    813.69    965.78
       db'      d#'                gb'       ab'       a#'
 b#        d5'       e5'    e#'        g5'        a5'       b5'  b#'
 -41.06   198.53    391.69 462.36     701.96     895.11  1088.27 1158.94
 
Keyboard 2: G-B#, 1/4-comma (Yamaha "key of E") with modified "5" notes
Keyboard 1: Eb-G#, 1/4-comma (Yamaha "key of C")

     76.04     310.26          579.47     772.63     1006.84  
      c#'        eb'             f#'        g#'         bb'
 _76.0|117.1_117.1|76.0_     _76.0|117.1_76.0|117.1_117.1|76.0_    
 c'         d'         e'    f'         g'         a'         b'   c''
 0        193.16    386.31 503.42    696.58      889.74   1082.89 1200
    193.16     193.16   117.11   193.16    193.16    193.16   117.11 

It may be noted that in this 1/4-comma meantone implementation (with five notes modified to provide pure fifths), accidental pairs such as c#/db and d#/eb are a lesser diesis (128:125 or about 41.06 cents) apart. In Vicentino's ideal scheme with five equal dieses to a whole-tone, these accidentals would be separated by 1/5-tone, or half of his minor semitone of 2/5-tone -- a "diesis" equal to one step in 31-tet, or about 38.71 cents.

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4. Concluding observations
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While Vicentino's full 31-tet system is required to obtain all of his genera and intervals, even a 19-note or 24-note implementation can reveal some interesting possibilities. For example, between db and e# on the second keyboard (Vicentino's third rank), we find an interval of about 345.25 cents in 1/4-comma tuning -- or 348.39 cents in 31-tet (9/31 octave).

Vicentino himself describes this interval as having an effect somewhere between a minor and a major third, and states its "irrational ratio" as approximately 5 1/2:4 1/2 -- or, converting to an integer ratio, 11:9. He finds this a concordant and useful vertical interval on the archicembalo, with its nature tending somewhat toward the major third, a sonorous consonance lending a certain kindred character to such an intermediate form.

In addition to being of interest from the viewpoint of Vicentino's own theory and practice, a 19-note or 24-note subset of his archicembalo can be also be helpful in playing some of the most adventurous works of composers such as Gesualdo, who goes as far as E# and B#.

For Renaissance and early 17th-century music which goes beyond the range of a 12-note meantone keyboard, but not quite this far, a more moderate "split key" arrangement of two manuals might be easier because it permits more unisons between the keyboards. Certain chromatic works of Orlando di Lasso and Gesualdo, for example, might have a range of Eb-A# or Ab-D# or the like, requiring only a couple of nonunisonal notes between the two keyboards (e.g. Eb/D# and Bb/A#; or Ab/G# and Eb/D#). Implementing this kind of arrangement for synthesizer(s) might only require selecting two transpositions of a preset 1/4-comma meantone option.

Incidentally, if one wants a closed Renaissance meantone system with 19 or 24 notes per octave, then 19-tet (almost identical to 1/3-comma meantone) is a logical choice. Here, one might choose 19-tet plus five extra notes to provide pure fifths below the virtually pure minor thirds of this tuning.[5]

1. This article by Bill Alves, originally appearing in 1/1: Journal of the Just Intonation Network 5(No.2):8-13 (Spring 1989), is available [here]. It includes a very helpful bibliography of material such as the extensive studies of Henry Kaufmann. [back to text]

2. Nicola Vicentino, Ancient Music Adapted to Modern Practice, tr. Maria Rika Maniates, ed. Claude V. Palisca (New Haven: Yale University Press, 1996). [back to text]

3. One point of ambiguity is that Vicentino describes the keys of the sixth rank as a "comma" higher than corresponding keys of the first rank, a term which can have various meanings in his usage. Here it seems reasonable to take the comma in question as the difference between a 31-tet or 1/4-comma meantone fifth and a pure 3:2 fifth -- in other words, actually 1/4 of a syntonic comma. This reading nicely fits in with Vicentino's description of the comma, cited by Alves, as an interval which can "help a consonance." Alternatively, a "comma" for Vicentino can also mean an interval of half an enharmonic diesis or 31-tet step -- roughly 19 cents. Since adding a comma in this sense to the usual tempered fifth of his system would yield a fifth of about 716 cents, or 14 cents wider than just, the first interpretation seems to me more likely. [back to text]

4. Such a custom synthesizer retuning of the five notes on the second keyboard supplying pure fifths would obviously be easier using a computer-based program such as the brilliant Scala by Manuel Op de Coul, or setting the tuning once on the synthesizer's front panel and then saving it to a RAM cartridge or the like for repeated use. [back to text]

5. Such pure fifths for 19-tet or 1/3-comma meantone are located a bit less conveniently than in 31-tet or 1/4-comma meantone from the viewpoint of usual Renaissance practice. In the latter case, for example, b5' (about 5.38 cents above b' in 1/4-comma, 5.18 cents in 31-tet) serves as a pure fifth for the common sonority E-G#-B, often sounded on the final of E Phrygian. In contrast, the tone we might call b-5' in 19-tet (about 7.17 cents below b' in 1/3-comma, or 7.22 cents in 19-tet) supplies the lowest tone of the somewhat unusual sonority B-D#-F#, outside the most typical 16th-century meantone range (Eb-G#) although it occurs in various adventurous works. The pure sonorities supported by the other "just fifth" keys in 1/3-comma or 19-tet are more prevalent: d-5' (D-F#-A); e-5' (E-G#-B); g-5' (G-B-D); and a-5' (A-C#-E). [back to text]

Most respectfully,

Margo Schulter
mschulter@value.net